d.5.1 scenarios to be included in hlc

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 764014

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D.5.1 – Scenarios to be included in HLC

Project acronym:

Project full title:

Call identifier:

Type of action:

Start date:

End date:

Project number:

SEA-TITAN

SEA-TITAN: Surging Energy Absorption Through Increasing Thrust And

efficiency

H2020-LCE-2017-RES-RIA-TwoStage

RIA

01/04/2018

31/03/2021

764014

WP5:

Due date:

Submission date:

Responsible partner:

Version:

Status:

Author(s):

Reviewer(s):

Deliverable type:

Dissemination level:

Laboratory testing of the novel PTO prototype

31/12/2018

25/02/2019

WAVEC

1.0

Final

Marco Alves

Aleix María Arenas

R: Report

PU: Public


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Version history

Version Date Author Revised Partner Description 1.0 23/02/2019 Marco Alves Aleix María Arenas Wedge Final

Statement of originality

This deliverable contains original unpublished work except where clearly indicated otherwise.

Acknowledgement of previously published material and of the work of others has been made

through appropriate citation, quotation or both.


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TABLE OF CONTENTS

1. EXECUTIVE SUMMARY .......................................................................................................... 4

2. TIME-DOMAIN MODEL DESCRIPTION ................................................................................... 5

3.1 Wave-to-Wire model ........................................................................................................... 5

3. TIME DOMAIN RESULTS ...................................................................................................... 10

4.1 Regular wave results ......................................................................................................... 12

4.2 Irregular wave results ........................................................................................................ 12

4.2.1 Centipod ..................................................................................................................... 13

4.2.2 CorPower .................................................................................................................... 19

4.2.3 SeaCap ........................................................................................................................ 21

4.2.4 Wedge ........................................................................................................................ 23

4. CONCLUSIONS ..................................................................................................................... 27

REFERENCES ................................................................................................................................ 28


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1. EXECUTIVE SUMMARY

This report presents a preliminary study on the dynamics and performance of the four wave

energy converters (WECs) under analysis in the Sea-Titan project. The four WECs are the

Centipod, CorPower, SeaCap and Wedge. The devices are depicted in Figure 1.1. The study is

based on the application of a tailor-made wave-to-wire model, developed to specifically

simulate the performance of these concepts. The wave-to-wire model essentially simulates the

chain of energy conversion from the hydrodynamic interaction between the wave energy

converter and the ocean waves to the power produced. An overview of the wave-to-wire model

and the model assumptions and simplifications adopted are discussed in Section 2.

The power take-off system implemented in the time-domain model here presented is a linear

reciprocating electrical generator. Due to the modular nature of the structure of the wave-to-

wire model, it could be relatively straightforward to implement modifications to the modelling

methodology including some modifications in the devices geometry, more complicated PTO

mechanisms and PTO control strategies.

To execute the model a full frequency evaluation of the device with a 3D difraction/radiation

code (WAMIT) was performed in advance. Its results were presented in D2.1 of the Sea-Titan

project. The output frequency dependent coefficients, the added mass, the hydrodynamic

damping and the complex amplitudes of the excitation force are required as inputs to the wave-

to-wire model.

Further data obtained with the wave-to-wire model includes time series of the most relevant

dynamic parameters that characterize the performance of the concepts. Namely, the

displacements and velocities of the WECs bodies, and the total power generated by the device

at the generator. These time-series results are presented in Section 3.


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2. TIME-DOMAIN MODEL DESCRIPTION

The dynamics of the four wave energy converters (WEC) under analysis in the Sea-Titan project

will be analyzed using the wave-to-wire model (W2W) or so-called time domain code developed

by WavEC. Wave-to-wire modelling tools allow for the modelling of the entire chain of energy

conversion from the hydrodynamic wave/device interaction to the power production. Although

the application of wave-to-wire models can be time demanding (typically more than frequency-

domain models but much less than CFD simulations), it allows the PTO dynamics to be described

in considerable detail and the overall performance of the devices to be calculated for any sea

state (fully described by spectral-density functions). For this project the tailor-made wave-to-

wire model will be applied to assess the overall performance of the WECs by obtaining time-

series of the motions and instantaneous power. It is then possible to generate power matrices

(that allows later to assess the annual power production for deployment locations characterized

by scatter diagrams), along with other relevant performance parameters such as

excursions/velocities, structural loads, and instantaneous power generation under the WECs

operational regime, etc. The time domain analysis begins with the application of WAMIT and

will follow with the application of the wave-to-wire time domain model and the developed post-

processing tool to generate power matrices for each of proposed the devices. WAMIT is a

numerical 3D radiation-diffraction boundary element method (BEM) panel model based on the

classic linear water wave theory and potential flow and is used to compute the complex

hydrodynamic coefficients required to be input into the wave-to-wire model. An overview of the

wave-to-wire model, outlining the general assumptions used in the implementation of the

numerical wave-to-wire model are given in the following Section 2.1. Furthermore, some

simplifications have been made to the modelling description in order to reduce complexity of

the system that would need to be modelled in the wave-to-wire model. A reduced number of

rigid body degrees of freedom are considered along with a simplified interpretation of the

geometry of the devices.

3.1 Wave-to-Wire model

This section presents a discussion on the general theory of building numerical post processing

tools (with respect to WAMIT or equivalent code outputs, see WAMIT User Manual) to model

the dynamics of WECs using a time domain approach. Overall, the developed Wave-to-Wire

model is based on Newton’s second law, which states that the inertial force is balanced by the

whole forces acting over the WEC’s captor. The entire force may be decomposed into

hydrodynamic and external sources. The hydrodynamic source comprises the excitation force,

inflicted by the incident waves; the buoyancy force, due to the variation in submergence caused

by the oscillatory motions of the device; and the radiation force, related to the pressure on the

device wetted surface, due to the fluid moved by the device oscillations. In addition, the device


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motion may also be constrained by external forces induced by both the PTO equipment and any

required mooring system. A convenient technique to characterize the radiation force in the time

domain is the state-space method, which involves representing the convolution integral of the

radiation hydrodynamic interaction by a small number of first order differential equations with

constant coefficients. Two different methodologies can be used to derive the coefficients of the

differential equations: i) explicitly from the specified impulse response function or ii) directly

from the frequency dependent coefficient or transfer function (computed by standard

hydrodynamic radiation-diffraction codes). Due to their expediency for numerical computation

two general methods are used: i) the first one consists of approximating the impulse response

function by a combination of exponential functions in the time domain and ii) the second one

approximates the transfer function by a complex rational function. Concerning the estimate of

the resultant excitation force on the captor: the methodology adopted consists of computing

the coefficients of both the free surface elevation and the excitation force (which are obtained

using standard radiation-diffraction codes for unitary amplitude incident waves). The free

surface coefficients, or more precisely the complex amplitudes of the free surface elevation, are

determined through a stochastic analysis based on a predefined wave spectrum (Pierson-

Moskowitz, JONSWAP, Bretschneider, etc). The frequency dependent excitation force for the

specified sea state is then obtained by multiplying the free surface coefficients and the excitation

force coefficients for unitary amplitude incident waves. Finally, the time dependent excitation

force results from the inverse Fourier transform of the frequency dependent excitation force.

To complete the WEC hydrodynamic modelling the hydrostatic force is also included, described

by a linear function of the WEC’s displacement, dependent on the device’s water-plane area.

Such an assumption is valid if the device motions are small, as expected within the devices

operational regime. The mooring dynamics can be neglected as the dynamics of the anchoring

cables has no significant memory effects. Thus, the mooring force, if required, can be modelled

by an additional non-linear restoring force. If the WEC’s displacement is small, the restoring

force due to the mooring system may also be linearized. In contrast, the PTO is typically

characterized by strong nonlinearities due to its stroke and force limitations and, usually, it

comprises important dynamics, due to, for instance, the inertia of a motor or a turbine or the

complexity of the control strategy to adjust the PTO load in accordance to the sea state. In the

case of the present work the PTO equipment consists of a reciprocating linear generator. Finally,

a global state-space realization is implemented, combining the hydrodynamic (linear) and non-

linear components (such as the dynamics of the PTO equipment or/and the mooring system) to

set the complete ODE system, which describes the entire dynamics of the device. A solution to

the ODE system is found by applying a solver based on an explicit Runge-Kutta (4,5) method, the

Dormand-Prince pair. Essentially, it is a one-step solver in computing y(tn), requiring only the

solution at the immediately preceding time point, y(tn−1).

To summarize, a schematic diagram of the required model inputs, the methodology followed,

and the outputs obtained from the wave-to-wire model is given in Fig. 2.1. In brief, the inputs of

the model are:

• Geometric parameters of the WEC;


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• Parameters describing the PTO and mooring system;

• Hydrodynamic coefficients (added mass and damping) and the excitation coefficients

(Computed with standard radiation/diffraction codes such as WAMIT);

• Wave spectrum.

The main outcomes of model include time-series data of:

• Motion/velocities of the captor/s;

• Loads on the captor/s;

• PTO loads;

• The instantaneous power produced;

The instantaneous power time-series may be averaged over time in order to calculate the mean

power produced for a given sea state. Multiple calculations over a series of sea states can be

computed in order to generate a power matrix for the concept under consideration. Table 2.1

outlines the inputs that are required for the wave-to-wire model and Table 2.2. outlines the

general assumptions used in the implementation of the numerical wave-to-wire model.

Figure 2.1 A scheme of the input required, methodology followed, and main output obtained from the wave-

to-wire model


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Table 2.1 A table of the general wave-to-wire model inputs.


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Table 2.2 A table of the general wave-to-wire model assumptions.


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3. TIME DOMAIN RESULTS

In this section results from the wave-to-wire modelling of the four WECs are presented. This

includes a set of demonstrative simulations results for irregular waves, which are shown for

three particular sea states. Three cases of the linear generator PTO operation are considered:

• No PTO, freely moving devices

• A PTO whose action is defined by the product of the velocity with a constant damping

coefficient

• A sub-optimum PTO whose action follows an optimum force.

For the sub-optimum PTO, the optimum force is given in the frequency domain by:

𝐹𝑃𝑇𝑂,𝑜𝑝𝑡(𝜔) = 𝐹𝑒(𝜔)𝑍∗(𝜔)

2𝐵(𝜔) + 2|𝑍(𝜔)|2

𝑅𝐿

where 𝐹𝑒 is the excitation force, 𝑍 is the hydrodynamic impedance, 𝐵is the hydrodynamic

damping, and 𝑅𝐿 is a factor representing the PTO losses. In this case, it was assumed no losses

of the PTO, i.e., 𝑅𝐿 = ∞. Therefore, the general expression above is simplified to:

𝐹𝑃𝑇𝑂,𝑜𝑝𝑡(𝜔) = 𝐹𝑒(𝜔)𝑍∗(𝜔)

2𝐵(𝜔)

In the current implementation the PTO force is assumed to be constrained, and therefore it is

called as sub-optimum PTO. For these exemplifying cases the limitation of the PTO force has

the value of 200 kN. With respect to the displacement and velocity, no constraint was imposed

in these simulations.

Before proceeding to the time-series results of irregular waves simulations, some regular wave

simulations were carried out, as brief consistency checks of the numerical wave-to-wire model.

It is assumed that unidirectional waves with zero degrees of incidence are considered here and

the most important rigid body mode is the heave mode, since the power capture depends on

the floaters vertical displacement. Moreover, a proper mooring system should constrain

substantially the motion in the other modes. The water depth is not the same for all the WECs,

they were set as indicated by the developers; the hydrodynamic coefficients presented in D2.1

of the Sea-Titan project were computed accordingly.

Table 3.1 Water depth considered for each WEC, as indicated by the developers.

WEC Centipod CorPower SeaCap WEDGE

Water Depth (m) 100 40 40 100


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Figure 3.1 Wedge geometry

Figure 3.2 CorPower geometry

Figure 3.3 Centipod geometry

Figure 3.4 SeaCap geometry


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4.1 Regular wave results

Simple tests were conducted in regular waves in order to understand if the model is providing

predictable responses in some particular conditions. Specifically, it is known that for long periods

the heaving bodies follow the free-surface elevation. Figure 3.5 shows the response of the four

technologies to regular waves with a period of 30s and 1m of amplitude. As it can be seen, the

bodies motions of all concepts follow the motion of the waves, as it was expected. Actually this

is true except for the particular case of the Centipod: since there are three floaters placed along

the propagation of the wave, only the one in the middle follows the free surface elevation, which

is measured at the position of the center floater; regarding the other two, one is slightly ahead

and the other is slightly behind the free surface elevation, due to the difference in the excitation

force’s phase.

Figure 3.5 Displacement of the all the devices, for a regular wave with a period of 30s and 1m amplitude.

4.2 Irregular wave results

For each technology and for each PTO case, simulations in irregular waves were considered with

three sea states with a total simulation of 1800s. It was assumed the Pierson-Moskowitz spectral

model, for the sea states defined by the pairs (TP, HS) shown in Table 3.


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Table 3.2 Sea states considered in the analysis.

Sea state 1 Sea state 2 Sea state 3

Peak Period (s) 6 9 13

Significant Wave Height (m) 1.5 3 5

It is important to note that the useful power absorbed is extremely dependent on the control

strategy (i.e. the force that the generator imposes on the bodies motion), so it is advised to use

the model to improve the control strategy to ultimately improve the WECs efficiency.

4.2.1 Centipod

The case of no PTO yields a maximum of the displacement between 4.4m and 5m (depending

on the floater) for the most energetic sea state, with RMS between 1.26m and 1.3m.

Figure 3.6 Displacement and velocity of the first body of the Centipod device, for the three states, with no

PTO modelling.


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Figure 3.7 Displacement and velocity of the second body of the Centipod device, for the three states, with

no PTO modelling.

Figure 3.8 Displacement and velocity of the third body of the Centipod device, for the three states, with no

PTO modelling.


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With a constant PTO damping coefficient, the mean power obtained, averaging for the three

floaters, was 22kW, 42kW and 58kW. In these simulations the maximum displacement was

between 4.4m and 4.8m, with a RMS between 1.18m and 1.23m.

Figure 3.9 Displacement, velocity and power of the first body of the Centipod device, for the three states,

with constant PTO damping coefficient.


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Figure 3.10 Displacement, velocity and power of the second body of the Centipod device, for the three

states, with constant PTO damping coefficient.

Figure 3.11 Displacement, velocity and power of the third body of the Centipod device, for the three

states, with constant PTO damping coefficient.


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Finally, for the sub-optimum PTO case, the mean power for the three sea states are 24kW,

63kW and 78kW, averaging for the three floaters. The maximum of the displacement for the

most energetic sea state is around 3.8m, with a RMS of 1.12m.

Figure 3.12 Displacement, velocity and power of the first body of the Centipod device, for the three states,

with sub-optimum PTO.

Table 3.3 Mean power for the different tested sea states (SS) and displacement at the most energetic sea

state (SS3) for the Centipod device. The displacement values presented are referring to the floater with the

largest displacement.

PTO operation scenario Mean Power (kW) Displacement at SS3

SS1 SS2 SS3 RMS (m) Max (m)

No PTO NA NA NA 1.30 5.0

PTO with constant damping coefficient

22 42 58 1.23 4.8

Sub-optimum PTO 24 63 78 1.12 3.8


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Figure 3.13 Displacement, velocity and power of the second body of the Centipod device, for the three

states, with sub-optimum PTO.

Figure 3.14 Displacement, velocity and power of the third body of the Centipod device, for the three

states, with sub-optimum PTO.


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4.2.2 CorPower

Looking to the CorPower WEC, and in particular to the maximum of the displacement, the freely

moving case for the most energetic se state (TP=13s, HS=5m) presents a value of about 4.5m,

with a root-mean-square (RMS) value of 1.3m.

For the case of a constant PTO damping coefficient, the maximum of the displacement is about

3.8m for the most energetic sea state, with a RMS of approximately 1.2m. The mean power at

the generator is 13kW, 40kW and 60kW, respectively for the lower, medium and higher energy

sea states.

Results obtained for the sub-optimum PTO case present a maximum of the displacement of

around 4.2m, with a RMS of 1.27m, for the most energetic sea state. The mean power is for the

three sea states of 48kW, 90kW and 100kW.

Figure 3.15 Displacement and velocity of the CorPower device, for three sea states, with no PTO modelling.


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Figure 3.16 Displacement, velocity and power of the CorPower device, for three sea states, with constant

PTO damping coefficient.


state (SS3) for the CorPower device.





13 40 60 1.20 3.8

Sub-optimum PTO 48 90 100 1.27 4.2


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Figure 3.17 Displacement, velocity and power of the CorPower device, for three sea states, with sub-

optimum PTO.

4.2.3 SeaCap

The simulations with no PTO resulted, for the most energetic sea state, in a maximum of the

displacement of approximately 4.5m, with a RMS of 1.27m.

In the case of the constant PTO damping coefficient, the displacement reaches a maximum of

4.4m, with a RMS of 1.25m, for the most energetic sea state. Regarding the mean power for the

three sea states, the results is 10kW, 29kW and 51kW.

With the sub-optimum PTO, the mean power obtained for the three sea states is 28kW, 48kW

and 63kW. In this case the maximum of the displacement is 3.7m, with the RMS of about 1.23m.


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Figure 3.18 Displacement and velocity of the SeaCap device, for three sea states, with no PTO modelling.

Figure 3.19 Displacement, velocity and power of the SeaCap device, for three sea states, with constant

PTO damping coefficient.


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Figure 3.20 Displacement, velocity and power of the SeaCap device, for three sea states, with sub-

optimum PTO.


state (SS3) for the SeaCap device.





10 29 51 1.25 4.4

Sub-optimum PTO 28 48 63 1.23 3.7

4.2.4 Wedge

The Wedge WEC principle is based on the relative motion between two heaving bodies, a top

floater and the spar below. For the cases with active PTO, besides the results for the motions of

each of the two bodies and the system instantaneous power, results are also shown for the

relative velocity.

For the case of no PTO, the maximum displacement obtained in the simulation was 5m for the

top floater and 3.6m for the body below, with the respective RMS of 1.44m and 1.25m.

In the case of the PTO with a constant damping coefficient, the resultant mean power is 12kW,

44kW and 60kW for the three sea states. The displacements of most energetic sea state reached

4.7m and 4.2m, for the top floater and the bottom component.


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The mean power obtained with the sub-optimum PTO was 76kW, 84kW and 163kW. The

maximum displacement for the most energetic sea state is 4.4m for the floater and around 30m

for the bottom body, with the RMS of 1.56m and 8.8m, respectively for the top and the bottom

bodies.

Figure 3.21 Displacement and velocity of the Wedge device, for three sea states, with no PTO modelling.


state (SS3) for the Wedge device. The displacement is given in the format (value for the floater / value for

the bottom body).



No PTO NA NA NA 1.44 / 1.25 5.0 / 3.6


12 44 60 1.48 / 1.16 4.7 / 4.2

Sub-optimum PTO 76 84 163 1.56 / 8.8 4.4 / 30.0


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Figure 3.22 Displacement and velocity of the Wedge device, for three sea states, with constant PTO

damping coefficient.

Figure 3.23 Relative velocity and power of the Wedge device, for three sea states, with constant PTO

damping coefficient.


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Figure 3.24 Displacement and velocity of the Wedge device, for three sea states, with sub-optimum PTO.

Figure 3.25 Relative velocity and power of the Wedge device, for three sea states, with sub-optimum PTO.


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4. CONCLUSIONS

In this document a Wave-to-Wire model to simulate the dynamics and power performance of

the four WECs under study in the Sea-Titan project was presented. The time-domain numerical

model was briefly described, focusing on its main components. Several time-series results on

regular and irregular waves were shown, namely the displacement and velocity of the WECs,

as well as the instantaneous power at the generator. This set of results illustrates some

features of the model, which besides the motions and the power, may also compute the

hydrodynamic, PTO and mooring loads.

Despite presenting only results for a limited number of sea states, it is possible to apply the

Wave-to-Wire model to a significant range of sea states, in order to build a power matrix.

Therefore, together with the information of the scatter diagram of a specific site, we may then

estimate the annual energy production.


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d.5.1 scenarios to be included in hlc

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